Download Angles Formed by Parallel Lines

2.2
Angles Formed by Parallel Lines
YOU WILL NEED
GOAL
• compass
Prove properties of angles formed by parallel lines and a
transversal, and use these properties to solve problems.
• protractor
• ruler
INVESTIGATE the Math
Briony likes to use parallel lines in her art. To ensure that she draws the
parallel lines accurately, she uses a straight edge and a compass.
?
A.
How can Briony use a straight edge and a compass to ensure
that the lines she draws really are parallel?
Draw the first line. Place a point, labelled P, above the line. P will be
a point in a parallel line.
P
B.
Draw a line through P, intersecting the first line at Q.
EXPLORE…
• Parallelbarsareusedin
therapytohelppeople
recoverfrominjuriestotheir
legsorspine.Howcouldthe
manufacturerensurethatthe
barsareactuallyparallel?
P
Q
C.
Using a compass, construct an
arc that is centred at Q and passes
through both lines. Label the
intersection points R and S.
P
R
Q
S
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2.2 Angles Formed by Parallel Lines
73
D.
Draw another arc, centred at P,
with the same radius as arc RS.
Label the intersection point T.
T
P
R
Q
S
E.
Draw a third arc, with centre T
and radius RS, that intersects
the arc you drew in step D.
Label the point of intersection
W.
T
P
W
R
Q
S
Communication
Tip
F.
IflinesPW and QSare
parallel,youcanrepresent
therelationshipusingthe
symbol i :
PW i QS
Draw the line that passes
through P and W. Show
that PW i QS.
T
P
W
R
Q
S
Reflecting
74
G.
How is /SQR related to /WPT ?
H.
Explain why the compass technique you used ensures that the two
lines you drew are parallel.
I.
Are there any other pairs of equal angles in your construction? Explain.
Chapter 2 Properties of Angles and Triangles
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APPLY the Math
example
1
Reasoning about conjectures involving
angles formed by transversals
Make a conjecture that involves the interior angles formed by parallel lines
and a transversal. Prove your conjecture.
Tuyet’s Solution
My conjecture: When a transversal intersects a pair of parallel lines, the
alternate interior angles are equal.
alternate interior angles
Twonon-adjacentinteriorangles
onoppositesidesofatransversal.
1
3 4
5
2
Statement
/1 5 /2
Justification
Corresponding
angles
/1 5 /3
Vertically
opposite
angles
Transitive
property
My conjecture is proved.
/3 5 /2
Idrewtwoparallellinesanda
transversalasshown,andInumbered
theangles.Ineedtoshowthat
/3 5 /2.
SinceIknow that thelinesare
parallel,thecorrespondingangles
areequal.
Whentwolinesintersect,theopposite
anglesareequal.
/2and/3arebothequalto/1,
so /2and/3areequaltoeach
other.
Ali’s Solution
My conjecture: When a transversal intersects a pair of parallel lines, the
interior angles on the same side of the transversal are supplementary.
1
3 4
5
2
/1 5 /2
/2 1 /5 5 180°
NEL
Ineedtoshowthat/3and/5are
supplementary.
Sincethelinesareparallel,the
correspondinganglesareequal.
Theseanglesformastraightline,
sotheyaresupplementary.
2.2 Angles Formed by Parallel Lines
75
Since/2 5 /1,Icouldsubstitute
/1for/2intheequation.
/1 1 /5 5 180°
/1 5 /3
alternate exterior angles
Verticallyoppositeanglesare
equal.Since/1 5 /3,Icould
substitute/3for/1inthe
equation.
/3 1 /5 5 180°
Twoexterioranglesformed
betweentwolinesanda
transversal,onoppositesides
ofthetransversal.
My conjecture is proved.
Your Turn
Naveen made the following conjecture: “ Alternate exterior angles are
equal.” Prove Naveen’s conjecture.
example
2
Using reasoning to determine unknown angles
Determine the measures of a, b, c, and d.
b
a
c
d
110°
Kebeh’s Solution
/a 5 110°
The110°angleand/aarecorresponding.Sincethe
linesareparallel,the110°angleand/aareequal.
/a 5 /b
/b 5 110°
Verticallyoppositeanglesareequal.
/c 1 /a 5 180°
/c 1 110° 5 180°
/c 5 70°
b � 110°
a � 110°
c � 70°
110°
/c 5 /d
/d 5 70°
The measures of the angles are:
/a 5 110°; /b 5 110°;
/c 5 70°; /d 5 70°.
d
/cand/aareinterioranglesonthesame
sideofatransversal.Sincethelinesare
parallel,/cand/aaresupplementary.
Iupdatedthediagram.
/cand/darealternateinteriorangles.Sincethe
linesareparallel,/cand/dareequal.
Your Turn
a)
Describe a different strategy you could use to determine the measure of /b.
b) Describe a different strategy you could use to determine the measure of /d.
76
Chapter 2 Properties of Angles and Triangles
NEL
example
3
Using angle properties to prove that lines are parallel
One side of a cellphone tower will be built as shown. Use the angle
measures to prove that braces CG, BF, and AE are parallel.
D
C
78°
78°
35°
78°
35°
22°
B
A
H
78°
22°
G
F
E
Morteza’s Solution: Using corresponding angles
/BAE 5 78° and /DCG 5 78°
Given
AE i CG
Whencorrespondinganglesareequal,thelines
areparallel.
/CGH 5 78° and /BFG 5 78°
Given
CG i BF
Whencorrespondinganglesareequal,thelines
areparallel.
AE i CG and CG i BF
SinceAEandBFarebothparalleltoCG,allthree
linesareparalleltoeachother.
The three braces are parallel.
Jennifer’s Solution: Using alternate interior angles
Statement
Justification
/CGB 5 35° and /GBF 5 35° Given
Alternate interior angles
CG i BF
Whenalternateinterioranglesare
equal,thelinesareparallel.
/FBE 5 22° and /BEA 5 22°
BF i AE
Given
Alternate interior angles
Whenalternateinterioranglesare
equal,thelinesareparallel.
CG i BF and BF i AE
Transitive property
The three braces are parallel.
SinceCGandAEarebothparalleltoBF,
theymustalsobeparalleltoeachother.
Your Turn
Use a different strategy to prove that CG, BF, and AE are parallel.
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2.2 Angles Formed by Parallel Lines
77
In Summary
Key Idea
• Whenatransversalintersectstwoparallellines,
i)thecorrespondinganglesareequal.
ii)thealternateinterioranglesareequal.
iii)thealternateexterioranglesareequal.
iv) theinterioranglesonthesamesideofthetransversalaresupplementary.
b
a c
d
i) a  e, b  f
c  g, d  h
ii) c  f, d  e
e g
f h
iii) a  h, b  g
iv) c  e  180°
d  f  180°
Need to Know
• Ifatransversalintersectstwolinessuchthat
i) thecorrespondinganglesareequal,or
ii) thealternateinterioranglesareequal,or
iii) thealternateexterioranglesareequal,or
iv) theinterioranglesonthesamesideofthetransversalare supplementary,
thenthelinesareparallel.
CHECK Your Understanding
1. Determine the measures of /WYD, /YDA, /DEB, and /EFS.
Give your reasoning for each measure.
K
L
M
C
115°
A
W
Y
N
B 80°
E
D
P
Q
R
45°
X
Z
F
S
2. For each diagram, decide if the given angle measures prove that the
blue lines are parallel. Justify your decisions.
a)
101°
101°
78
Chapter 2 Properties of Angles and Triangles
b)
c)
51°
119°
d)
73°
73°
85°
85°
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PRACTISING
3. A shelving unit is built with two pairs of parallel
planks. Explain why each of the following
statements is true.
a) /k 5 /p
e) /b 5 /m
b) /a 5 /j
f) /e 5 /p
c) /j 5 /q
g) /n 5 /d
d) /g 5 /d
h) /f 1 /k 5 180°
4. Determine the measures of the indicated angles.
a)
x
b)
55°
a b
e
c)
d
a
e
f
48°
112°
c b
e
f
d
g h
j k
n o
d
w
y 120°
c
f
l m
p q
c
g
b
a
5. Construct an isosceles trapezoid. Explain your method of construction.
6. a) Construct parallelogram SHOE, where /S 5 50°.
b) Show that the opposite angles of parallelogram SHOE are equal.
7. a) Identify pairs of parallel lines and transversals in the embroidery
pattern.
b) How could a pattern maker use the properties of the angles created
by parallel lines and a transversal to draw an embroidery pattern
accurately?
8. a) Joshua made the following conjecture: “If AB ' BC and
BC ' CD, then AB ' CD.” Identify the error in his reasoning.
Joshua’s Proof
Statement
Justification
AB ' BC
Given
BC ' CD
Given
Transitive property
AB ' CD
b) Make a correct conjecture about perpendicular lines.
Embroideryhasarichhistory
intheUkrainianculture.In
onestyleofembroidery,called
nabiruvannia,thestitchesare
madeparalleltothehorizontal
threadsofthefabric.
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2.2 Angles Formed by Parallel Lines
79
9. The Bank of China tower in Hong Kong was the tallest building in
Asia at the time of its completion in 1990. Explain how someone in
Hong Kong could use angle measures to determine if the diagonal
trusses are parallel.
10. Jason wrote the following proof.
Identify his errors, and correct his proof.
Given:
Q
QP ' QR
P
QR ' RS
QR i PS
R
Prove: QPSR is a parallelogram.
TheBankofChinatowerhasa
distinctive3-Dshape.Thebase
ofthelowerpartofthebuilding
isaquadrilateral.Thebaseof
thetopisatriangle,makingit
stableandwind-resistant.
S
Jason’s Proof
Statement
Justification
/PQR 5 90° and /QRS 5 90° Lines that are
perpendicular meet at
right angles.
QP i RS
Since the interior angles
on the same side of a
transversal are equal, QP
and RS are parallel.
QR i PS
QPSR is a parallelogram
Given
QPSR has two pairs of
parallel sides.
11. The roof of St. Ann’s Academy
in Victoria, British Columbia,
has dormer windows as shown.
Explain how knowledge of parallel
lines and transversals helped the
builders ensure that the frames for
the windows are parallel.
12. Given:
Prove:
^FOX is isosceles.
/FOX 5 /FRS
/FXO 5 /FPQ
PQ i SR and SR i XO
13. a) Draw a triangle. Construct a line
X
P
S
F
Q
R
segment that joins two sides of
your triangle and is parallel to the
third side.
b) Prove that the two triangles in your construction are similar.
80
Chapter 2 Properties of Angles and Triangles
O
NEL
14. The top surface of this lap
120°
harp is an isosceles trapezoid.
a) Determine the measures
of the unknown angles.
b) Make a conjecture about
the angles in an isosceles
trapezoid.
x
z
y
15. Determine the measures of all the
unknown angles in this diagram,
given PQ i RS.
R
29°
Q
T
54°
P
48°
S
16. Given AB i DE and DE i FG, show that
/ACD 5 /BAC 1 /CDE.
B
A
C
G
F
E
D
17. When a ball is shot into the side or end of a pool table, it will rebound
off the side or end at the same angle that it hit (assuming that there is
no spin on the ball).
a) Predict how the straight paths of the ball will compare with each
other.
b) Draw a scale diagram of the top of a pool
table that measures 4 ft by 8 ft. Construct the
B
trajectory of a ball that is hit from point A on
one end toward point B on a side, then C, D,
and so on.
A
c) How does path AB compare with path CD ?
How does path BC compare with path DE ? Was
your prediction correct?
d) Will this pattern continue? Explain.
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C
D
2.2 Angles Formed by Parallel Lines
81
18. Given:
Prove:
QP i SR
RT bisects /QRS.
QU bisects /PQR.
QU i RT
Q
T
S
P
U
R
Closing
19. a) Ashley wants to prove that LM i QR. To do this, she claims that
she must show all of the following statements to be true:
i) /LCD 5 /CDR
X
ii) /XCM 5 /CDR
C
iii) /MCD 1 /CDR 5 180°
Do you agree or disagree? Explain. L
D
b) Can Ashley show that the lines are
parallel in other ways? If so, list
Q
these ways.
M
R
Y
Extending
20. Solve for x.
a)
(3x  10)°
b)
(6x  14)°
(9x  32)°
(11x  8)°
21. The window surface of the large pyramid at Edmonton City Hall is
composed of congruent rhombuses.
a) Describe how you
could determine the
angle at the peak of the
pyramid using a single
measurement without
climbing the pyramid.
b) Prove that your strategy
is valid.
TheEdmontonCityHallpyramidsareacity
landmark.
82
Chapter 2 Properties of Angles and Triangles
NEL
Applying Problem-Solving Strategies
Checkerboard Quadrilaterals
One strategy for solving a puzzle is to use inductive reasoning. Solve
similar but simpler puzzles first, then look for patterns in your solutions
that may help you solve the original, more difficult puzzle.
The Puzzle
How many quadrilaterals can you count on an 8-by-8 checkerboard?
The Strategy
A.
B.
C.
How many squares or rectangles can you count on
a 1-by-1 checkerboard?
Draw a 2-by-2 checkerboard. Count the quadrilaterals.
Draw a 3-by-3 checkerboard, and count the quadrilaterals.
Develop a strategy you could use to determine the number
of quadrilaterals on any checkerboard. Test your strategy on
a 4-by-4 checkerboard.
E. Was your strategy effective? Modify your strategy if necessary.
F. Determine the number of quadrilaterals
on an 8-by-8 checkerboard. Describe your
strategy.
G. Compare your results and strategy with the
results and strategies of your classmates.
Did all the strategies result in the same
solution? How many different strategies
were used?
H. Which strategy do you like the best?
Explain.
D.
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2.2 Angles Formed by Parallel Lines
83